Warm Up The following table lists the 50 states.

Transcription

1 .notebook Warm Up The following table lists the 50 states. (a) Obtain a simple random sample of size 10 using Table I in Appendix A, a graphing calculator, or computer software. 4 basic sampling techniques simple random sampling stratified sampling systematic sampling cluster sampling 1

2 .notebook Obtaining a stratified sampling A stratified sample is obtained by separating the population into nonoverlapping groups called strata and then obtaining a simple random sample from each stratum. The individuals within each stratum should be homogeneous (or similar) in some way. Obtaining a stratified sampling With this population, we can easily separate the individuals by color. 2

3 .notebook Obtaining a stratified sampling Once we have the strata determined, we need to decide how many individuals to select from each stratum. The number selected should be proportional. In our case, 1/4 of the individuals in the population are blue, so 1/4 of the sample should be blue as well. Working things out, we can see that a stratified (by color) random sample of 4 should have 1 blue, 1 green, and 2 reds. Obtaining a stratified sampling Problem: The president of DePaul University wants to conduct a survey to determine the community s opinion regarding campus safety. The president divides the DePaul community into three groups: resident students, nonresident (commuting) students, and staff (including faculty) so that he can obtain a stratified sample. Suppose there are 6,204 resident students, 13,304 nonresident students, and 2,401 staff, for a total of 21,909 individuals in the population. The president wants to obtain a sample of size 100, with the number of individuals selected from each stratum weighted by the population size. Calculate the numbers of resident students, nonresident students, and staff that will make up the sample. 3

4 .notebook Obtaining a systematic sampling A systematic sample is obtained by selecting every kth individual from the population. The first individual selected corresponds to a random number between 1 and k. The idea behind obtaining a systematic sample is relatively simple: Select a number k, (we find k by taking N/n and rounding down to the nearest integer) randomly select a number between 1 and k, survey that individual, then survey every kth individual thereafter. For example, we might select to survey every k = 8th individual. We randomly select a number between 1 and 8 such as 5. This means we survey the 5th, = 13th, = 21st and so on, until we reach the desired sample size. Obtaining a systematic sampling In general we find k by taking N/n and rounding down to the nearest integer. For the above population (N), we want to use for k in order to obtain a sample size (n) of 4? For our starting point, we pick a random number between 1 and k. If we randomly pick the number 2, the sample would be made up of 2,,, and. 4

5 .notebook Obtaining a systematic sampling Systematic sampling works well when the individuals are already lined up in order. In the past, students have often used this method when asked to survey a random sample of students. Since we don't have access to the complete list, just stand at a corner and pick every 10th person walking by. Obtaining a systematic sampling If the size of the population is unknown, there is no mathematical way to determine k. It must be chosen by determining a value of k that is not so large that we are unable to achieve our desired sample size, but not so small that we do not obtain a sample size that is representative of the population. Problem: The manager of Kroger Food Stores wants to measure the satisfaction of the store s customers. List the steps that would go into the design of a systematic sampling technique that can be used to obtain a sample of 15 customers. 5

6 .notebook Warm Up Without looking at your notes, write some details that you remember about systematic and stratified sampling. 6

7 .notebook Obtaining a cluster sampling A cluster sample is obtained by selecting all individuals within a randomly selected collection or group of individuals. In essence, we use cluster sampling when the population is already broken up into groups (clusters), and each cluster represents the population. That way, we just select a certain number of clusters. Obtaining a cluster sampling If we want a random sample of size four, we just select two of the clusters. We would number the clusters 1 6 and use technology to randomly select two random numbers. It might look something like this: 7

8 .notebook Obtaining a cluster sampling Example Suppose your company makes light bulbs, and you'd like to test the effectiveness of the packaging. You don't have a complete list, so simple random sampling doesn't apply, and the bulbs are already in boxes, so you can't order them to use systematic. And all the bulbs are essentially the same, so there aren't any characteristics with which to stratify them. To use cluster sampling, a quality control inspector might select a certain number of entire boxes of bulbs and test each bulb within those boxes. In this case, the boxes are the clusters. Obtaining a cluster sampling The mathematics department at a university wishes to administer a survey to a sample of students taking college algebra.the department is offering 32 sections of college algebra, similar in class size and makeup, with a total of 1,280 students. They would like the sample size to be roughly 10% of the population of college algebra students this semester. How might the department use cluster sampling to obtain the desired results? 8

9 .notebook In general, are you satisfied or dissatisfied with the way things are going in the country? I would like to obtain responses to this question from a sample of 8 students in this class. How might that be done using... a) simple random sampling? c)systematic sampling? b) stratified sampling? d)cluster sampling? Convenience sampling A convenience sample is a sample in which the individuals are easily obtained and not based on randomness. Unfortunately, they're rarely representative. Think of the radio call in show, those people in the shopping malls trying to survey you about your purchasing habits, or even the voting on American Idol! Why would those examples be considered non representative of the population? 9

10 .notebook Convenience sampling The most popular types of convenience sampling are those in which the individuals in the sample are self selected (the individuals themselves decide to participate in a survey). These are also called voluntary response samples. For example, Dateline will present a story regarding a certain topic and ask its viewers to tell us what you think by completing a questionnaire online. This is a poor design because the individuals who decide to be in the sample generally have strong opinions about the topic. Why is this a poor design? A more typical individual in the population will not bother logging on to a computer to complete a survey. Any inference made regarding the population from this type of sample should be made with extreme caution. "Graffiti's the most passionate literature there is, you know? It's always like 'Bush sucks!'... 'U2 Rocks!'... I want to make indifferent graffiti. 'Toy Story 2 was okay!'... 'I like Sheryl as a friend, but I'm not sure about taking things further'... 'This is a bridge!' Demetri Martin 10

11 .notebook Multistage sampling Obtaining samples using a combination of the sampling techniques. As an example of multistage sampling, consider Nielsen Media Research. Nielsen randomly selects households and monitors the television programs these households are watching through a "People Meter". The meter is an electronic box placed on each TV within the household.the "People Meter" measures what program is being watched and who is watching it. Nielsen selects the households with the use of a two stage sampling process. Stage 1: Using U.S. Census data, Nielsen divides the country into geographic areas. The areas are typically city blocks in urban areas and geographic regions in rural areas. About 6,000 areas are randomly selected. Stage 1 is an example of which sampling method? Stage 2: Nielsen sends representatives to the selected areas and lists the households within the area. The households are then randomly selected. This is an example of which sampling method? 11